Card shuffling and the decomposition of tensor products

Abstract

Let H be a subgroup of a .nite group G. We use Markov chains to quantify how large r should be so that the decomposition of the r tensor power of the representation of G on cosets on H behaves (after renormalization) like the regular representation of G. For the case where G is a symmetric group and H a parabolic subgroup, we .nd that this question is precisely equivalent to the question of how large r should be so that r iterations of a shu.ing method randomise the Robinson-Schensted-Knuth shape of a permutation. This equivalence is remarkable, if only because the representation theory problem is related to a reversible Markov chain on the set of representations of the symmetric group, whereas the card shu.ing problem is related to a nonreversible Markov chain on the symmetric group. The equivalence is also useful, and results on card shu.ing can be applied to yield sharp results about the decomposition of tensor powers.